HAL Id: jpa-00247276
https://hal.archives-ouvertes.fr/jpa-00247276
Submitted on 1 Jan 1996
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The Electrodynamics of Low Dimensional Metals
G. Grüner
To cite this version:
G. Grüner. The Electrodynamics of Low Dimensional Metals. Journal de Physique I, EDP Sciences, 1996, 6 (12), pp.1711-1718. �10.1051/jp1:1996168�. �jpa-00247276�
The Electrodynamics of Low Dimensional Metals
G. Grüner (*)
Department of Physics, University of California, Los Angeles, Los Angeles, CA 90095-1547, USA
(Received 3 June 1996, received in final form 15 July 1996, accepted 29 August 1996)
PACS.78.20.-e Optical properties of bulk materials and thin films PACS.75.30.Fv Spin-density waves
PACS.71.45.Lr Charge-density-wave systems
Abstract. I discuss the unusual electrodynarnics observed in the metallic state of various low dimensional metals which at low temperatures develop charge density wave (CDW) and spin density wave (SDW) ground states. In ail cases, we observe a low energy mode with a small
spectral weight and
a finite energy excitation. These results suggest a highly unusual metallic
state. The findings will be contrasted with the theoretical models of low dimensional metals.
1. Introduction
Research, with the goal of obtaining organic metals bas started at least in the eyes of this au- thor with trie systematic investigation of trie physical properties of TCNQ (tetracyanoquin- odimetan) salts by Igor Schegolev and bis group. Soon after trie results were summarized in a
review article in Physica Status Soiidi [1j, metallic TCNQ salts and soon other organic metals
were synthesized.
With trie discovery of superconducting and other broken symmetry ground states, trie at- tention bas focussed on trie properties of these ground states. In this short review I will, however, discuss trie unusual features of trie metallic state of several materials with a hnear
chain structure; trie subject bas become, agam, trie focus of attention.
It bas been recognized early that fluctuation eifects play an important rote above trie three- dimensional ordering transition T3D in quasi one-dimensional systems. Trie mean-field solution of an ideal one-dimensional system leads to a finite transition temperature T~~ where long
range order develops and trie system undergoes a Peierls transition to a charge-density-wave (CDW) spin-density-wave (SDW) or superconducting (SC) ground state. This, however, is
an artifact of trie mean-field approximation which neglects trie rote played by fluctuations of trie order parameter. These are important in low dimensions, and a strictly one-dimensional
system with only short range interactions does net develop long range order at finite tempera-
ture. Real quasi one-dimensional materials, however, are highly amsotropic three-dimensional
systems with interchain electronic Coulomb interactions and tunneling leading to coupling of the fluctuations which develop along each chain. This coupling results m a finite transition
temperature T3D below which three-dimensional long range order occurs. For weak interchain
(*) e-mail: grunerslphysics.ucla.edu
© Les Éditions de Physique 1996
1712 JOURNAL DE PHYSIQUE I N°12
coupling, T3D is significantly smaller than T~~ The region below T~~ is characterized by one-
dimensional fluctuations which, at some temperature T* > T3D, cross over to fluctuations with two- or three-dimensional character. T* is trie temperature at which trie transverse correlation
length fi becomes comparable to trie interchain spacing. These eifects bave been shown to be
important years ago. Recently, there is renewed interest in trie behavior of highly anisotropic conductors in their metallic state. Trie reason for this is twofold. First, various theories based
on trie one-dimensional (1D) Hubbard model, and other models predict a non-Fermi hquid
state with profound implications on the spectroscopic signatures in the metallic state. Trie
general name Luttinger liqmd is used to describe the new quantum liquid state which follows from the diiferent solutions of1D interacting electron systems. Second, a variety of experi-
ments conducted on linear chain compounds with highly anisotropic electronic properties give
indications of unusual features which might be signatures of such novel states of solids. As real materials such as the linear-chain compounds are not strictly one-dimensional; this raises
interesting questions about the extent to which strictly 1D models are applicable to experi-
ments performed on actual materials. In addition, the interplay between the periodicity of the
electronic correlations (given by the Fermi wavevector kF and the lattice periodicity may lead to unusual features such as discommensurations and nonlinear charge and spm excitations.
The experiments optical measurements in a broad spectral range which I am going to summarize, have been obtained on materials with charge density wave and spin density wave ground states. In the inorganic compounds Ko.3Moo3 and (TaSe4)21 a CDW state develops at 180 K and 260 K, while in the organic sait (TMTSF)2PF6 a SDW state is observed below 12 K. In the organic compound TTF-TCNQ, which will be mentioned at the end of this review,
a sequence of phase transitions to various CDW state starts below 59 K.
2. Experimental Techniques
Various techniques have been employed to obtain the electrodynamic response in an extremely
wide frequency range. In the microwave range, a cavity perturbation technique [2j pioneered
m Igor Schegolev's group [3j was used to measure the surface impedance Ès
" Rs + iXs at
various temperatures. By placing a needle shaped crystal in the maximum of the electric field
of a cyhndrical cavity and measuring the change m width (Ar) and central frequency (A f)
of the resonance, it is possible to calculate both the surface resistance Rs and the surface
reactance Xsi
~~ ~°~Î( ~~~ °~~ ~°ÎÎ
' ~~
where Zo " 47r/c = 4.19 x 10~~° s/cm is the free space impedance. The resonator constant (
can be calculated from the geometry of the cavity and the sample.
The complex conductivity (à
= ai + ia2) and permittivity (ê = fi + ie2) can be calculated from the surface impedance usmg Ès
" Zo uc/2iâ and ê = 1+21â /vc where v is the resonance
frequency. In addition, the absorptivity A can be calculated using the relation
A
= 1 R
= ~(~ 1 + ~)~ + ~~~~~~ , (2)
o o
~
o
~
where R is the reflectivity. In the submillimeter wave spectral range a coherent source spec-
trometer utilizing Backward Wave Oscillator was employed.
In the optical range from 14 cm~~ up to 10~ cm~~, standard polarized reflection experiments
(E parallel and perpendicular to the chains) were performed by usmg four diiferent spectrom-
eters with overlapping frequency ranges. In the mfrared spectral range two Fourier transform
mterferometer were used with a gold mirror as a reference.
10°
Ko_~MOO~
T = 300 K
."
~ ,Z ."
t E 1 b " ,./ E I b
/j 10~~
,:" 1'
cL ;.'
;.'
~
;.' /
D Cavity Pert.
l' . Reflectivity
10'~ ,)~~ ÎÎ~~~~~~~~~
10~ 10~
Frequency (cm~~)
Fig. l. Frequency dependence of the room temperature absorptivity of Ko 3Mo03 in bath orien- tations E
ii b and E 1b. The open squares
were obtained by measuring the surface resistance, the sohd circles represent data of quasi-optical reflectivity measurements. The solid fines show the optical reflectivity data. The dashed fines show the results of the dispersion analysis of the data. We did not try to describe the large number of phonon features which can be seen in the perpendicular direction.
3. Results
In Figures 1 and 2, 1display the optical reflectivity and conductivity [4j of Ko 3Moo3 in two diiferent directions (one with parallel, one with perpendicular) of polarization to the chain axis b.
The electrodynamic response of Ko 3Mo03 is dominated by a substantial amsotropy con-
nected with the one-dimensional nature of the charge transport. The conductivity perpendic-
ular to the b direction, ail, is frequency mdependent below 2000 cm~~ at
room temperature.
We find no dispersion in the sub-mm frequency spectra of e and a, and ai(T
= 300 K) m
5Ç (Qcm)~~, approximately equal to its de value, corresponding to the Hagen-Rubens behavior
seen in Figure 1. The E i b excitation spectra, however, show two overall features: a structure at 3000 cm~~, with
a minimum in the infrared spectral range, and an enhanced conductivity
at low frequencies. Assuming single partiale transport for the parallel direction, a
= ne~T/mb,
with n
= 5.4 x 10~~ cm~3 the electron density and mb
" o.9mo the band mass leads to
~ = (27rcT)~~ = 4500 cm~~, implying a frequency independent conductivity weÎÎ below ~. In contrast, we see a decreasing with decreasing frequency for v < looo cm~~ Lowering the tem- perature enhances this feature. Below the transition the temperature dependent conductivity
shows an activated behavior below TCDv/ adc
" ao exp(-A/kBT), with A/hc m 700 cm~~,
m agreement with Îow-temperature optical data; both giving evidence for a weÎÎ defined CDW gap m the ordered state. This value is mdicated in Figure 2 by the open arrow. We conclude therefore that the feature seen around 3000 cm~~ is a pseudogap due to the correlations which exist in the metallic state [8].
At low frequencies we find a pronounced enhancement of ajj below 50 cm~~ This implies an
additional contribution to the conductivity m the Ko.3Moo3 spectra for E ii b with characteristic frequency v < 10 cm~l We beheve, that the low frequency response is due to trie fluctuating
1714 JOURNAL DE PHYSIQUE I N°12
10~ K0.3MOO~ ~DcVaiue8
° Cavity Pen,
j300K (~jitY Pert. 200
Î
-
E I
FîtÎOÎ~)°~~~~~~~
~,
Fit 200 K
£ 103
~
Î
cÎ
/
É ioz
E 1 b
./
10'
lo~
Frequency (cm~~)
Fig. 2. Frequency dependence of the conductivity of Ko 3Mo03 for E ii b (T
= 300 K, 200 K) and
for E 1b (300 K). The solid points show the result of a direct measurement of the conductivity by the
Fabry-Perot reflectivity technique, and the open squares are calculated from the surface impedance
measurements. The dashed fines represent the dispersion analysis of the entire data set. The open
arrow indicates the single particle gap 2Ap. The solid arrows show the de conductivities.
CDW response which bas two important features: finite de conductivity and a peak at 3 cm~~.
Trie latter appears at trie same frequency where trie pinned CDW occurs well below TCDv/, and we argue that it anses in trie fluctuation region as well due to trie interaction of trie fluctuation CDW segments with impurities. In contrast to what happens in trie ordered state, the fluctuating CDW segments cari contribute to the de response due to thermal fluctuations
over trie impurity mduced barriers. The feature shown in Figure 2 bas been modeled by a
harmonic oscillator and a Drude response [4]. In trie absence of firm theoretical predictions it is not possible to decide whether trie low frequency contribution to trie conductivity is due to a
single response of trie collective mode, or arises due to two separate processes, with both smgle partiale (due to uncondensed electrons) and collective (due to condensed electrons) response being important.
Trie optical properties of trie organic hnear chain compound (TMTSF)2PF6 are also highly anisotropic, as is evident from Figures 3 and 4. For polarization parallel to trie chain direction,
the optical response has two components: a zero energy mode and a finite energy excitation, centered around 200 cm~~ The combined spectral weight of trie two components
la)~(uJ)dur + a)~(uJ) du = ~~~ = ~~ (3) mÎ
leads to a total plasma frequency uJp/(27rc) = 1.1 x 10~ cm~~ which, within experimental error,
is independent of temperature (inset of Fig. 4a). This value is m full agreement with trie plasma frequency derived from the known electron concentration n
= 1.24 x 10~~ cm~3 and
a
bandmass mb * me. As trie mset of Figure 4a also demonstrates, only approximately 1$l of trie total spectral weight is in trie zero energy mode. This value can be calculated in two ways.
First, trie contribution of trie finite energy mode can be subtracted from trie conductivity, and the remaining zero energy mode can be mtegrated; this gives uJ)~/(27rc) m 1.1 x 103 cm~~
j~ ;" à-1' -
~10~~ _JÀ~ y."° ii ----
_:°~ ~p" ~<' O .
" _;
,~» O .
,'"
o ~ ô~'
j 10
~ E
,.~ (b)
10~~
~~~(",,~
_~
" ;
~~ l
Fig. 3. Absorptivity A
= 1- R of a (TMTSF)2PF6 single crystal as a function of frequency
at several temperatures for both polarization directions, (a) parallel to the chains (E
ii a) and (b)
perpendicular ta the chains (E
ii b'). The open symbols are microwave and millimeter wave data,
the sohd symbols are results obtained with the submilhmeter wave spectrometer. The solid fines
are standard optical reflectivity measurements, and the dashed fines indicate the input used for the
Kramers-hronig analysis.
Second, the zero crossing of the dielectric constant near 2j cm~~, when corrected foi trie higher
frequency contributions from trie finite energy mode, gives a plasma frequency of approximately
x lo3 cm~~, at all temperatures where this mode is clearly defined. It is also seen from trie inset of Figure 4a that this spectral weight is independent of temperature.
Trie zero energy feature cannot be described by a simple Drude response, but with a fre- quency dependent mass and relaxation rate, m*(uJ) and r*(uJ):
~~
me (r*)2 + (uJm*/me)2' ~~~
where m*(uJ) and r* (uJ) are related by trie Kramers-Kronig relation. Trie implications of these
frequency dependances are not clear at this time.
Trie feature around 200 cm~~ becomes sharper with decreasmg temperature. Trie overall shape trie peaked structure with a pronounced asymmetry closely resembles trie spectrum of a one-dimensional semiconductor with a significant broadening, added to a flat background.
Simple arguments advanced for a semiconductor with a 1D density of states (and frequency independent transition probability) lead to a square root smgularity at the gap Eg, and
~~~~ "
huJ
Eg]~~/~ ÎÎ
Î~. ~~~
Essentially trie same structure follows from trie mean field solution of the density wave ground
state with zero collective mode spectral weight, and from the solution of the 1D Hubbard mortel for half filling. The overall shape found experimentally is similar to that given by equation là),
1716 JOURNAL DE PHYSIQUE I N°12
20000
j~~ (TMTSF)zPF6
E I a
1Î
~
~
_ f
'~ Î
Îà
Î
CÎ Q
1
ioo
~ 300
1
(cm~~)
1000 (TaSe4)zI
T = 270K
~£
i
C~ T
o
ooo
85K
o
1
(cm~~
Fig. 5. Frequency dependent conductivity in typical low dimensional materials. The single partiale
gaps as evaluated from the temperature dependance of the de conductivities measured well below the 3D transition
are also indicated on the Figure.
For (TMTSF)2PF6 the explanation of the unusual electrodynamics may be diiferent. There is substantial evidence that magnetic correlations are important in the TMTSF salts. At the same
time, electron counting arguments together with the observed dimerization give an electron concentration of one electron per unit cell. For a strictly one dimensional band, the material at T
= o should be an msulator m the presence of Coulomb interactions for which the Hubbard mortel provides an appropriate description. The metallic state observed by experiment is then due to deviations from the strictly one-dimensional half-filled band case with finite temperature eifects conceivably also playmg an important role. Calculations based on the lD Hubbard model with electron concentration close to half-filling do not lead to a simple incommensurate SDW with a well defined gap at T
= o and semiconducting behavior, but to a situation m which commensurate regions where the electron concentration corresponds to half-filling are separated by discommensurations, leading to the appropriate overall electron concentration.
The spectroscopic signatures of such a state bave been calculated [10,11] and include a limite energy absorption together with a translationally invariant zero energy mode. Trie limite energy absorption is due to single particle-like excitations of the half-filled regions (with features similar
1718 JOURNAL DE PHYSIQUE I N°12
to those obtained for the half-filled Hubbard mortel) and near to half-filling the zero frequency
mode is due to the translationally invariant response of discommensurations. Near to half-
filling, the spectral weight of the finite energy excitations gives the dominant contribution to the total spectral weight with the discommensurations (with their number approaching zero as
the material approaches half-filling) having a small spectral weight, a feature also found here experimentally.
While diflerent mechamsms may be operative in the diiferent materials discussed here, look-
ing at Figure 5 it is diflicult to escape the conclusion that the fundamental features, a low energy mode and a finite energy excitation are the common characteristics of low dimensional metallic systems with highly anisotropic band structure. The question, which of these features
are due to single particle and which are due to collective excitations, has only partially been resolved to-date.
Acknowledgments
The experimental results discussed here bave been obtained by A. Schwartz and S. Donovan in close collaboration with L. Degiorgi, M. Dressel and B. Gorchounov. Trie samples were grown by B. Alavi. I am also grateful to L. Gor'kov for discussmg trie results.
The experimental results bave been reported earlier, together with trie implication of trie
findings and comparison with other experimental results. Consequently, trie references below list only those publications which the interested reader should consult for details.
References
Ill Schefolev I.F., Phys. Status Sohdi 12 (1972) 9.
[2] Klein O., Donovan S., Dressel M. and Griiner G., Int. J. Infrared Miihmeter Waves 14
(1993) 2324; Donovan S., Klein O., Dressel M., Holczer K. and Griiner G., Int. J. Infrared
Miiiimeter Waves 14 (1993) 2359; Dressel M., Klein O., Donovan S. and Griiner G., Inn.
J. Infrared Miiiimeter Waves 14 (1993) 2389.
[3] Buravov L.I. and Schegolev I.F., Pub. i. Tekh. Eksp. 4 (.1971) 171.
[4] Gorchounov B-P-, Volkov A.A., Kozlov G-V-, Degiorgi L., Blank A., Csiba T., Dressel M.,
Kim Y., Schwartz A. and Griiner G., Phys. Rev. Lent. 73 (1994) 308.
[5] Schwartz A., Dressel M., Alavi B., Blank A., Dubois S. and Griiner G., Phys. Rev. B 52.
(1995) 5643.
[fil Degiorgi L. et ai., Phys. Rev. Lett. 76 (1996) 3838.
[7] Dressel M. et ai., Phys. Rev. Lett., to be published.
[8j Lee P-A-, Rice T.M. and Anderson P-W-, Soiid State Comm. 14 (1974) 7033. Griiner G.:
Density Waves in Solids (Addison Wesley, Reading, MA, 1994).
[9j Basista H., Bonn D.A-, Timusk T., Voit J., Jerome D. and Bechgaard K., Phys. Rev. B 42 (1990) 4088.
[loi Preuss R. et ai., Phys. Rev. Lent. 73 (1994) 732.
[11j Mon M., Fukuyama H. and Imada I., J. Phys. Soc. Jpn 63 (1994) 1639.
